It can be proved that the limit in (REF ) exists and is finite.
What is more, Ignat and Jerrard [1]}
gave a characterisation of \(W\)  in terms of the Green function
for the Laplace-Beltrami operator on \(M\) , thus extending
earlier results by Bethuel, Brezis and Hélein [2]}
in the Euclidean setting. Let \(\mathrm {Vol}(M) := \int _M\mathrm {vol}_g\) . Let
\(G\colon \lbrace (x, \, y)\in M\times M\colon x\ne y\rbrace \rightarrow \mathbb {R}\) 
